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On the blow-up scenario for some modified Serre-Green-Naghdi equations

Billel Guelmame

Abstract

The present paper deals with a modified Serre-Green-Naghdi (mSGN) system that has been introduced by Clamond et al. to improve the dispersion relation. We present a precise blow-up scenario of the mSGN equations and we prove the existence of a class of solutions that develop singularities in finite time. All the presented results hold also for the Serre-Green-Naghdi system with weak surface tension.

On the blow-up scenario for some modified Serre-Green-Naghdi equations

Abstract

The present paper deals with a modified Serre-Green-Naghdi (mSGN) system that has been introduced by Clamond et al. to improve the dispersion relation. We present a precise blow-up scenario of the mSGN equations and we prove the existence of a class of solutions that develop singularities in finite time. All the presented results hold also for the Serre-Green-Naghdi system with weak surface tension.
Paper Structure (11 sections, 11 theorems, 82 equations, 1 figure)

This paper contains 11 sections, 11 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The equations
  3. Other similar equations
  4. Outline
  5. Derivation
  6. Derivation
  7. The dispersion relation
  8. Main results
  9. Preliminaries
  10. Blow-up criteria
  11. Blow-up results

Key Result

Theorem 1

Let $\beta > 0$, $\bar{h}>0$ and $s \geqslant 2$, then, for any $(h_0-\bar{h},u_0) \in H^s(\mathds{R})$ satisfying $\inf_{x\, \in\, \mathds{R}}h_0(x)>0$ there exists $T>0$ and $(h-\bar{h},u) \in C^0([0,T],H^s(\mathds{R})) \cap C^1([0,T],H^{s-1}(\mathds{R}))$ a unique solution of SGN2 such that Moreover, the solution satisfies the conservation of the energy

Figures (1)

  • Figure 1: Characteristics.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 6 more